Neural operators of backstepping controller gain kernels for an ODE cascaded with a reaction-diffusion equation

In this paper, we consider the neural operators for PDE backstepping designs for an ODE cascaded with a reactiondiffusion equation. Through deep neural network approximation of nonlinear operators, commonly known as DeepONet, we demonstrate the continuity of the mapping from the plant PDE functional coefficient to the kernel PDE solutions, prove the existence of an arbitrarily close DeepONet approximation to the kernel PDEs, and establish that the DeepONet approximated gains guarantee stabilization when replacing the exact backstepping gain kernels. The numerical simulation illustrates that the DeepONet is two orders of magnitude faster than PDE solvers for such gain functions.